The theory of determinants in the historical order of development, by Sir Thomas Muir.

106 HISTORY OF THE THEORY OF DETERMINANTS

WILLIAMSON, B. (1872): BUCHWALD, E. (1872).
[Condition for a maximum or a minimum in a function of any
number of variables. Quart. Journ. of Math., xii. pp. 48-51;
or his Differential Calculus, 1PI edition, pp. 340-343, 2 n
edition, pp. 363-367, etc.]
[Question 3683. Educ. Times, xxiv. p. 296, xxv. p. 18; or Math.
from Educ. Times, xvii. pp. 66-68.]
[Betingelsen for at den algebraisk rationale homogene Function af
anden Grad af n variable er positiv for alle Vcerdier af de
variable eller negativ for dem alle. Tidsskrift for Math., (3)
ii. pp. 20-25.]
The real object here is to obtain the conditions that must be
fulfilled in order that a quadric may remain positive for all real
values of the variables.
The investigation, due in essence to Lagrange (1759),* is based
on a simple transformation, three examples of which we may write
for ourselves in the form
X?
al. 32CI = (a,x + a?J ) +( aCb2 f y2'
(1, = x1''20' a3z )2 + b a3
a,1 a a3 X I alb I 1 lb, I
b9 b', b3 ly I ac'I C(C3 IZ,
Ce C2 C3 z
c y I V z w
a 3 4 '(ax+- -+at4a) 2 + Y~9Jab1 ab
bI (bb a b a4 1 a1c2 Iac I 1 ja1c4j
b) b, b b ILC, al lc,~o' aleq I z
C1 C2 c3 c4 z K 1dI I ald3j al(d,.,Iw,
dii d" 3 djw
where, it must be borne in mind, the square arrays are axisymmetric.t
* See immediately preceding footnote.
t This restriction may be done away with if we alter the squared expressions on
the right hand into
(a x + a.,y) (alx + b,y), (alx + a.2y + a-z) (adx + bly + c~z),